| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s + 4·13-s + 2·15-s + 4·17-s + 21-s − 4·23-s − 25-s + 27-s + 8·31-s + 2·35-s + 6·37-s + 4·39-s − 12·41-s − 8·43-s + 2·45-s − 8·47-s + 49-s + 4·51-s + 6·53-s − 4·59-s + 12·61-s + 63-s + 8·65-s − 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.10·13-s + 0.516·15-s + 0.970·17-s + 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.43·31-s + 0.338·35-s + 0.986·37-s + 0.640·39-s − 1.87·41-s − 1.21·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s − 0.520·59-s + 1.53·61-s + 0.125·63-s + 0.992·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.646426408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.646426408\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.61932558995776, −14.16312839281405, −13.77597433947825, −13.28923905186634, −12.97187997810142, −12.03979226856815, −11.74147826264766, −11.12315462189600, −10.32587143201686, −9.942887403428159, −9.692496612220164, −8.826232290560643, −8.270164147227912, −8.095356957989199, −7.285698718700645, −6.445329986093547, −6.222739822016729, −5.432089314031874, −4.943170305360723, −4.117718962164052, −3.506949265595598, −2.922808956413121, −2.029670168206110, −1.595941803711284, −0.7787596649608472,
0.7787596649608472, 1.595941803711284, 2.029670168206110, 2.922808956413121, 3.506949265595598, 4.117718962164052, 4.943170305360723, 5.432089314031874, 6.222739822016729, 6.445329986093547, 7.285698718700645, 8.095356957989199, 8.270164147227912, 8.826232290560643, 9.692496612220164, 9.942887403428159, 10.32587143201686, 11.12315462189600, 11.74147826264766, 12.03979226856815, 12.97187997810142, 13.28923905186634, 13.77597433947825, 14.16312839281405, 14.61932558995776
